classical mechanics - Analytic proof of the non-integrability of the Henon-Heiles system?

The Henon-Heiles potential is

$$U(x,y ) = \frac{1}{2} (x^2 + y^2 + 2 x^2 y - \frac{2}{3} y^3) .$$

This is a two degree-of-freedom system. The full Hamiltonian is

$$H = p_x^2 + p_y^2 + U(x,y ) .$$

It is shown by numerics that it is non-integrable. But can one prove it rigorously analytically? The problem boils down to proving the non-existence of a second first-integral/integral-of-motion.

If this problem is too difficult, is there any simpler model whose non-integrability can be proven analytically?